Zigzag Persistence Research Articles

Harder–Narasimhan filtrations of persistence modules

AbstractThe Harder–Narasimhan (HN) type of a quiver representation is a discrete invariant parameterised by a real‐valued function (called a central charge) defined on the vertices of the quiver. In this paper, we investigate the strength and limitations of HN types for several families of quiver representations which arise in the study of persistence modules. We introduce the skyscraper invariant, which amalgamates the HN types along central charges supported at single vertices, and generalise the rank invariant from multiparameter persistence modules to arbitrary quiver representations. Our four main results are as follows: (1) We show that the skyscraper invariant is strictly finer than the rank invariant and incomparable to the generalised rank invariant; (2) we characterise the set of complete central charges for zigzag (and hence, ordinary) persistence modules; (3) we extend the preceding characterisation to rectangle‐decomposable multiparameter persistence modules of arbitrary dimension; and finally, (4) we show that although no single central charge is complete for interval‐decomposable ladder persistence modules, a finite set of central charges is complete.

Zigzag persistence for image processing: New software and applications

Topological image analysis is a powerful tool for understanding the structure and topology of images, being persistent homology one of its most popular methods. However, persistent homology requires a chain of inclusions of topological spaces, which can be challenging for digital images. In this article, we explore the use of zigzag persistence, a recent variant of traditional persistence, for digital image processing. To this end, new algorithms are developed to build a simplicial complex associated to a digital image and to compute the relationships between homology classes of a sequence of binary images via zigzag persistence. Additionally, we provide a simple software to use them. We demonstrate its effectiveness by applying it to a real-world problem of analyzing honey bee sperm videos.

Time-Aware Knowledge Representations of Dynamic Objects with Multidimensional Persistence

Learning time-evolving objects such as multivariate time series and dynamic networks requires the development of novel knowledge representation mechanisms and neural network architectures, which allow for capturing implicit time-dependent information contained in the data. Such information is typically not directly observed but plays a key role in the learning task performance. In turn, lack of time dimension in knowledge encoding mechanisms for time-dependent data leads to frequent model updates, poor learning performance, and, as a result, subpar decision-making. Here we propose a new approach to a time-aware knowledge representation mechanism that notably focuses on implicit time-dependent topological information along multiple geometric dimensions. In particular, we propose a new approach, named Temporal MultiPersistence (TMP), which produces multidimensional topological fingerprints of the data by using the existing single parameter topological summaries. The main idea behind TMP is to merge the two newest directions in topological representation learning, that is, multi-persistence which simultaneously describes data shape evolution along multiple key parameters, and zigzag persistence to enable us to extract the most salient data shape information over time. We derive theoretical guarantees of TMP vectorizations and show its utility, in application to forecasting on benchmark traffic flow, Ethereum blockchain, and electrocardiogram datasets, demonstrating the competitive performance, especially, in scenarios of limited data records. In addition, our TMP method improves the computational efficiency of the state-of-the-art multipersistence summaries up to 59.5 times.

𝐾-theory of multiparameter persistence modules: Additivity

Persistence modules stratify their underlying parameter space, a quality that makes persistence modules amenable to study via invariants of stratified spaces. In this article, we extend a result previously known only for one-parameter persistence modules to grid multiparameter persistence modules. Namely, we show the K K -theory of grid multiparameter persistence modules is additive over strata. This is true for both standard monotone multi-parameter persistence as well as multiparameter notions of zig-zag persistence. We compare our calculations for the specific group K 0 K_0 with the recent work of Botnan, Oppermann, and Oudot, highlighting and explaining the differences between our results through an explicit projection map between computed groups.

Harder-Narasimhan filtrations and zigzag persistence

We introduce a sheaf-theoretic stability condition for finite acyclic quivers. Our main result establishes that for representations of affine type A˜ quivers, there is a precise relationship between the associated Harder-Narasimhan filtration and the barcode of the periodic zigzag persistence module obtained by unwinding the underlying quiver.

Algebraic stability theorem for derived categories of zigzag persistence modules

We study distances on zigzag persistence modules from the viewpoint of derived categories and Auslander–Reiten quivers. The derived category of ordinary persistence modules is derived equivalent to that of arbitrary zigzag persistence modules, depending on a classical tilting module. Through this derived equivalence, we define and compute distances on the derived category of arbitrary zigzag persistence modules and prove an algebraic stability theorem. We also compare our distance with the distance for purely zigzag persistence modules introduced by Botnan–Lesnick and the sheaf-theoretic convolution distance due to Kashiwara–Schapira.

Zigzag persistence for coral reef resilience using a stochastic spatial model.

A complex interplay between species governs the evolution of spatial patterns in ecology. An open problem in the biological sciences is characterizing spatio-temporal data and understanding how changes at the local scale affect global dynamics/behaviour. Here, we extend a well-studied temporal mathematical model of coral reef dynamics to include stochastic and spatial interactions and generate data to study different ecological scenarios. We present descriptors to characterize patterns in heterogeneous spatio-temporal data surpassing spatially averaged measures. We apply these descriptors to simulated coral data and demonstrate the utility of two topological data analysis techniques-persistent homology and zigzag persistence-for characterizing mechanisms of reef resilience. We show that the introduction of local competition between species leads to the appearance of coral clusters in the reef. We use our analyses to distinguish temporal dynamics stemming from different initial configurations of coral, showing that the neighbourhood composition of coral sites determines their long-term survival. Using zigzag persistence, we determine which spatial configurations protect coral from extinction in different environments. Finally, we apply this toolkit of multi-scale methods to empirical coral reef data, which distinguish spatio-temporal reef dynamics in different locations, and demonstrate the applicability to a range of datasets.

Temporal network analysis using zigzag persistence

This work presents a framework for studying temporal networks using zigzag persistence, a tool from the field of Topological Data Analysis (TDA). The resulting approach is general and applicable to a wide variety of time-varying graphs. For example, these graphs may correspond to a system modeled as a network with edges whose weights are functions of time, or they may represent a time series of a complex dynamical system. We use simplicial complexes to represent snapshots of the temporal networks that can then be analyzed using zigzag persistence. We show two applications of our method to dynamic networks: an analysis of commuting trends on multiple temporal scales, e.g., daily and weekly, in the Great Britain transportation network, and the detection of periodic/chaotic transitions due to intermittency in dynamical systems represented by temporal ordinal partition networks. Our findings show that the resulting zero- and one-dimensional zigzag persistence diagrams can detect changes in the networks’ shapes that are missed by traditional connectivity and centrality graph statistics.

Algebraic stability theorem for derived categories of zigzag persistence modules

We study distances on zigzag persistence modules from the viewpoint of derived categories and Auslander–Reiten quivers. The derived category of ordinary persistence modules is derived equivalent to that of arbitrary zigzag persistence modules, depending on a classical tilting module. Through this derived equivalence, we define and compute distances on the derived category of arbitrary zigzag persistence modules and prove an algebraic stability theorem. We also compare our distance with the distance for purely zigzag persistence modules introduced by Botnan–Lesnick and the sheaf-theoretic convolution distance due to Kashiwara–Schapira.

Hyperplane restrictions of indecomposable $n$-dimensional persistence modules

Understanding the structure of indecomposable $n$-dimensional persistence modules is a difficult problem, yet is foundational for studying multipersistence. To this end, Buchet and Escolar showed that any finitely presented rectangular $(n-1)$-dimensional persistence module with finite support is a hyperplane restriction of an $n$-dimensional persistence module. We extend this result to the following: If $M$ is any finitely presented $(n-1)$-dimensional persistence module with finite support, then there exists an indecomposable $n$-dimensional persistence module $M'$ such that $M$ is the restriction of $M'$ to a hyperplane. We also show that any finite zigzag persistence module is the restriction of some indecomposable $3$-dimensional persistence module to a path.

Generalized persistence diagrams for persistence modules over posets

When a category $\mathcal{C}$ satisfies certain conditions, we define the notion of rank invariant for arbitrary poset-indexed functors $F:\mathbf{P} \rightarrow \mathcal{C}$ from a category theory perspective. This generalizes the standard notion of rank invariant as well as Patel's recent extension. Specifically, the barcode of any interval decomposable persistence modules $F:\mathbf{P} \rightarrow \mathbf{vec}$ of finite dimensional vector spaces can be extracted from the rank invariant by the principle of inclusion-exclusion. Generalizing this idea allows freedom of choosing the indexing poset $\mathbf{P}$ of $F: \mathbf{P} \rightarrow \mathcal{C}$ in defining Patel's generalized persistence diagram of $F$. Of particular importance is the fact that the generalized persistence diagram of $F$ is defined regardless of whether $F$ is interval decomposable or not. By specializing our idea to zigzag persistence modules, we also show that the barcode of a Reeb graph can be obtained in a purely set-theoretic setting without passing to the category of vector spaces. This leads to a promotion of Patel's semicontinuity theorem about type $\mathcal{A}$ persistence diagram to Lipschitz continuity theorem for the category of sets.

Using Zigzag Persistent Homology to Detect Hopf Bifurcations in Dynamical Systems

Bifurcations in dynamical systems characterize qualitative changes in the system behavior. Therefore, their detection is important because they can signal the transition from normal system operation to imminent failure. In an experimental setting, this transition could lead to incorrect data or damage to the entire experiment. While standard persistent homology has been used in this setting, it usually requires analyzing a collection of persistence diagrams, which in turn drives up the computational cost considerably. Using zigzag persistence, we can capture topological changes in the state space of the dynamical system in only one persistence diagram. Here, we present Bifurcations using ZigZag (BuZZ), a one-step method to study and detect bifurcations using zigzag persistence. The BuZZ method is successfully able to detect this type of behavior in two synthetic examples as well as an example dynamical system.

A torus model for optical flow

We propose a torus model for high-contrast patches of optical flow. Our model is derived from a database of ground-truth optical flow from the computer-generated video Sintel, collected by Butler et al. in A naturalistic open source movie for optical flow evaluation. Using persistent homology and zigzag persistence, popular tools from the field of computational topology, we show that the high-contrast 3 × 3 patches from this video are well-modeled by a torus, a nonlinear 2-dimensional manifold. Furthermore, we show that the optical flow torus model is naturally equipped with the structure of a fiber bundle, related to the statistics of range image patches.

The reflection distance between zigzag persistence modules

By invoking the reflection functors introduced by Bernstein et al. (Russ Math Surv 28(2):17–32, 1973), in this paper we define a metric on the space of all zigzag modules of a given length, which we call the reflection distance. We show that the reflection distance between two given zigzag modules of the same length is an upper bound for the $$\ell ^1$$ -bottleneck distance between their respective persistence diagrams.

Algebraic stability of zigzag persistence modules

The stability theorem for persistent homology is a central result in topological data analysis. While the original formulation of the result concerns the persistence barcodes of $\mathbb{R}$-valued functions, the result was later cast in a more general algebraic form, in the language of \emph{persistence modules} and \emph{interleavings}. In this paper, we establish an analogue of this algebraic stability theorem for zigzag persistence modules. To do so, we functorially extend each zigzag persistence module to a two-dimensional persistence module, and establish an algebraic stability theorem for these extensions. One part of our argument yields a stability result for free two-dimensional persistence modules. As an application of our main theorem, we strengthen a result of Bauer et al. on the stability of the persistent homology of Reeb graphs. Our main result also yields an alternative proof of the stability theorem for level set persistent homology of Carlsson et al.

Interval decomposition of infinite zigzag persistence modules

We show that every pointwise finite-dimensional infinite zigzag persistence module decomposes into a direct sum of interval persistence modules.

Zigzag persistent homology for processing neuronal images

We apply the ideas of zigzag persistence to determine the objects of interest in stacks of neuronal images, locating and marking different dendrites. In particular, this allows us to recognize some 3D properties of the objects, distinguishing dendrites that cross, but not intersect, in the ambient space. The algorithms are implemented in a Fiji/ImageJ plugin, usable on two different kinds of images.

Zigzag Zoology: Rips Zigzags for Homology Inference

For \(n\) points sampled near a compact set \(X\), the persistence barcode of the Rips filtration built from the sample contains information about the homology of \(X\) as long as \(X\) satisfies some geometric assumptions. The Rips filtration is prohibitively large; however, zigzag persistence can be used to keep the size linear in \(n\), with a constant factor depending only (exponentially) on the intrinsic dimension of \(X\). We present several species of Rips-like zigzags and compare them with respect to the signal-to-noise ratio, a measure of how well the underlying homology is represented in the persistence barcode relative to the noise in the barcode at the relevant scales. Some of these Rips-like zigzags have been available as part of the Dionysus library for several years while others are new. Interestingly, we show that some species of Rips zigzags will exhibit less noise than the (nonzigzag) Rips filtration itself. Thus, Rips zigzags can offer improvements in both size complexity and signal-to-noise ratio. Along the way, we develop new techniques for manipulating and comparing persistence barcodes from zigzag modules. In particular, we give methods for reversing arrows and removing spaces from a zigzag while controlling the changes occurring in its barcode. These techniques were developed to provide our theoretical analysis of the signal-to-noise ratio of Rips-like zigzags, but they are of independent interest as they apply to zigzag modules generally.

Zigzag Persistence

We describe a new methodology for studying persistence of topological features across a family of spaces or point-cloud data sets, called zigzag persistence. Building on classical results about quiver representations, zigzag persistence generalises the highly successful theory of persistent homology and addresses several situations which are not covered by that theory. In this paper we develop theoretical and algorithmic foundations with a view towards applications in topological statistics.